Narrow Results

In this paper, we investigate the existence, uniqueness, and asymptotic behaviors of mild solutions of parabolic evolution equations on complex plane, in which the diffusion operator has the form ${\overline{\square }}_{\phi }=\overline{D}{\overline{D}}^{\ast }$, where $\overline{D}f=\bar{\partial }f+{\phi }_{\bar{z}}f$, the function $\phi$ is smooth and subharmonic on $ℂ$, and ${\overline{D}}^{\ast }$ is the formal adjoint of $\overline{D}$. Our method combines certain estimates of heat kernel associating with the homogeneous linear equation of Raich [Heat equations in $ℝ\times ℂ$, J. Funct. Anal. 240(1) (2006) 1–35] and a fixed point argument.

Everyone knows about the complicated solution of the nonlinear Fredholm integro-differential equation in general. Hence, often, authors attempt to obtain the approximate solution. In this paper, a numerical method for the solutions of the nonlinear Fredholm integro-differential equation (NFIDE) of the second kind in the complex plane is presented. In fact, by using the properties of Rationalized Haar (RH) wavelet, we try to give the solution of the problem. So far, as we know, no study has yet been attempted for solving the NFIDE in the complex plane. For this purpose, we introduce the continuous integral operator and real valued function. The Banach fixed point theorem guarantees that, under certain assumptions, the integral operator has a unique solution. Furthermore, we give an upper bound for the error analysis. An algorithm is presented to compute and illustrate the solutions for some numerical examples.

This paper’s purpose is to study the nonlinear Fredholm implicit integro-differential equation in the complex plane, where the term implicit integro-differential means that the derivative of unknown function is founded inside of the integral operator. Initially, according to Banach fixed point theory, we ensure that the equation has a unique solution under particular conditions. However, we exhibit a numerical process based on the conjunction between Nyström and Picard methods, for the sake of approximating solutions of this equation. In addition to that, the convergence analysis of this numerical process is demonstrated, and some illustrated numerical examples are presented.

This paper is devoted to the analogue of Jackson-Bernstein classic theorem on curves in the complex plane in the integral metric. In the uniform metric this matter in the form of a problem was formulated by J. Walsh and was investigated in the works of S. Y. Alper, A. I. Markushevich, S. N. Mergelian, J. I. Mamedkhanov and others. In the uniform metric this problem is close to a terminal solution. The similar problem on the curves in a complex plane in the integral metric was not studied enough and in this formulation it is considered for the first time.